Dynamically modeling a hydraulic fracture

ABSTRACT

Techniques for determining a hydraulic fracture dimension include identifying pressure signal values of a first fracturing fluid in a monitor wellbore that represent a pressure change in the first fracturing fluid that is induced by formation of a second hydraulic fracture from a treatment wellbore; determining a particular pressure signal value of the plurality of pressure signal values; based on the determined particular pressure signal value, determining a particular dimension of the second hydraulic fracture formed from the treatment wellbore; and determining, based at least in part on (i) the determined particular dimension, and (ii) a first pressure signal value of the plurality of pressure signal values that is less than the determined particular pressure signal value, a first intermediate dimension of the second hydraulic fracture that is less than the determined particular dimension of the second hydraulic fracture.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority under 35 U.S.C. § 119 to U.S.Provisional Patent Application Ser. No. 62/536,210, filed on Jul. 24,2017, and entitled “DYNAMICALLY MODELING A HYDRAULIC FRACTURE,” theentire contents of which are hereby incorporated by reference.

TECHNICAL FIELD

This specification relates to systems and method for dynamicallymodeling one or more hydraulic fractures to adjust or control ahydraulic fracturing system.

BACKGROUND

Certain geologic formations, such as unconventional reservoirs in shale,sandstone, and other rock types, often exhibit increased hydrocarbonproduction subsequent to one or more completion operations beingperformed. One such completion operation may be a hydraulic fracturingoperation, in which a liquid is pumped into a wellbore to contact thegeologic formation and generate fractures throughout the formation dueto a pressure of the pumped liquid (e.g., that is greater than afracture pressure of the rock formation). In some cases, anunderstanding of a size or other characteristics of the generatedhydraulic fractures may be helpful in understanding a potentialhydrocarbon production from the geologic formation.

SUMMARY

In a general implementation according to the present disclosure, astructured data processing system for determining a hydraulic fracturedimension includes one or more hardware processors and a memory incommunication with the one or more hardware processors. The memorystores a data structure and an execution environment. The data structurestores data that includes a plurality of pressure signal values of afirst fracturing fluid in a monitor wellbore formed from a terraneansurface into a subsurface formation. The first fracturing fluid is indirect fluid communication with a first hydraulic fracture formed fromthe monitor wellbore into the subsurface formation. Each of theplurality of pressure signal values includes a pressure change in thefirst fracturing fluid that is induced by formation of a secondhydraulic fracture from a treatment wellbore in the subsurfaceformation. The second hydraulic fracture is formed by a secondfracturing fluid in the treatment wellbore. The execution environmentincludes a fracture growth solver, a user interface module, and atransmission module. The execution environment is configured to performoperations including determining a particular pressure signal value ofthe plurality of pressure signal values; based on the determinedparticular pressure signal value, determining a particular dimension ofthe second hydraulic fracture formed from the treatment wellbore; anddetermining, based at least in part on (i) the determined particulardimension, and (ii) a first pressure signal value of the plurality ofpressure signal values that is less than the determined particularpressure signal value, a first intermediate dimension of the secondhydraulic fracture that is less than the determined particular dimensionof the second hydraulic fracture. The user interface module generates auser interface that renders one or more graphical representations of thedetermined first intermediate dimension of the second hydraulicfracture. The a transmission module that transmits, over one or morecommunication protocols and to a computing device, data that representsthe one or more graphical representations.

In an aspect combinable with the general implementation, the particulardimension includes at least one of a half-length of the second hydraulicfracture, a height of the second hydraulic fracture, or an area of thesecond hydraulic fracture.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding identifying a first fracture stage group of the treatmentwellbore that includes a set of hydraulic fractures formed from thetreatment wellbore, the set of hydraulic fractures including the secondhydraulic fracture formed from the treatment wellbore.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding minimizing an error inequality that includes a ratio of thedetermined fluid pressure and the determined particular pressure signalvalue; and determining, based on the minimized error inequality, acommon dimension of each of the hydraulic fractures in the set ofhydraulic fractures formed from the treatment wellbore.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding determining, based on the minimized error inequality, adimension of the first hydraulic fracture and a dimension between thefirst hydraulic fracture and the set of hydraulic fractures formed fromthe treatment wellbore.

In another aspect combinable with any of the previous aspects, the errorinequality includes a penalty function.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding determining the particular dimension of the second hydraulicfracture formed from the treatment wellbore based at least in part on adeviation of the particular dimension from the common dimension of eachof the hydraulic fractures in the set of hydraulic fractures formed fromthe treatment wellbore.

In another aspect combinable with any of the previous aspects, theoperation of determining the particular dimension of the secondhydraulic fracture formed from the treatment wellbore further includesperturbing a plurality of values of the particular dimension of thesecond hydraulic fracture as a function of the dimension of the firsthydraulic fracture and the dimension between the first hydraulicfracture and the set of hydraulic fractures formed from the treatmentwellbore.

In another aspect combinable with any of the previous aspects, theoperation of perturbing the plurality of values of the particulardimension of the second hydraulic fracture includes determining anoptimal value of the particular dimension based on a numerical modelthat includes the plurality of values of the particular dimension of thesecond hydraulic fracture; the plurality of values of the dimension ofthe first hydraulic fracture; and the dimension between the firsthydraulic fracture and the set of hydraulic fractures formed from thetreatment wellbore.

In another aspect combinable with any of the previous aspects, theintermediate dimension includes at least one of an intermediatehalf-length of the second hydraulic fracture, an intermediate height ofthe second hydraulic fracture, or an intermediate area of the secondhydraulic fracture.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding determining, based at least in part on (i) the determinedparticular dimension, and (ii) a second pressure signal value of theplurality of pressure signal values that is less than the determinedparticular pressure signal value, a second intermediate dimension of thesecond hydraulic fracture that is less than the determined particulardimension of the second hydraulic fracture.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding generating a fracture growth curve of the second hydraulicfracture based on the first and second intermediate dimensions and thedetermined particular dimension.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding generating a fracture growth curve of the second hydraulicfracture based on a continuum that includes the first and secondintermediate dimensions and the determined particular dimension.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding determining additional intermediate dimensions of the secondhydraulic fracture based on the generated fracture growth curve.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding graphically presenting the generated fracture growth curve toa user.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding determining, based on the generated fracture growth curve, atleast one hydraulic fracturing operation action. The user interfacemodule is configured to generate a user interface that renders one ormore graphical representations of the at least one hydraulic fracturingoperation action. The transmission module is configured to transmit,over the one or more communication protocols and to the computingdevice, data that represents the one or more graphical representationsof the at least one hydraulic fracturing operation action.

In another aspect combinable with any of the previous aspects, the atleast one hydraulic fracturing operation action includes at least one ofan action that adjusts a viscosity of the second fracturing fluid pumpedto the treatment wellbore; an action that adjusts a proppantconcentration in the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a pumping rate of the second fracturingfluid pumped to the treatment wellbore; or an action that adjusts adiversion schedule for the treatment wellbore.

In another aspect combinable with any of the previous aspects, the atleast one hydraulic fracturing operation action includes at least one ofan action that adjusts a viscosity of a third fracturing fluid relativeto a viscosity of the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a proppant concentration of the thirdfracturing fluid relative to a proppant concentration of the secondfracturing fluid pumped to the treatment wellbore; an action thatadjusts a pumping rate of the third fracturing fluid relative to apumping rate of the second fracturing fluid pumped to the treatmentwellbore; or an action that adjusts a diversion schedule for a thirdwellbore relative to a diversion schedule for the treatment wellbore.

In another aspect combinable with any of the previous aspects, theparticular pressure signal value corresponds to at least one of ashut-in time instant of the treatment wellbore upon a cessation ofpumping of the second fracturing fluid into the treatment wellbore; or amaximum pressure of the second fracturing fluid pumped into thetreatment wellbore.

In another aspect combinable with any of the previous aspects, thefracture growth solver is configured to perform operations furtherincluding normalizing the dimension between the first hydraulic fractureand the set of hydraulic fractures formed from the treatment wellbore inthe numerical model to at least one dimensionless value that representsa dimension of the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore.

In another aspect combinable with any of the previous aspects, thedimensionless value is associated with an estimated aspect ratio of thesecond hydraulic fracture that includes a ratio of a height of thesecond hydraulic fracture to a half-length of the second hydraulicfracture, an estimated overlap percentage between the first hydraulicfracture and the second hydraulic fracture, and an estimated offsetbetween the first hydraulic fracture and the second hydraulic fracture.

In another aspect combinable with any of the previous aspects, thenumerical model includes an N-dimensional hypercube.

In another aspect combinable with any of the previous aspects, theN-dimensional hypercube includes a poromechanical model that describes aporomechanical interaction of the second hydraulic fracture and thefirst hydraulic fracture.

In another general implementation according to the present disclosure, acomputer-implemented method and non-transitory computer readable mediafor determining a hydraulic fracture dimension include identifying aplurality of pressure signal values of a first fracturing fluid in amonitor wellbore, formed from a terranean surface into a subsurfaceformation, the first fracturing fluid in direct fluid communication witha first hydraulic fracture formed from the monitor wellbore into thesubsurface formation, each of the plurality of pressure signal valuesincluding a pressure change in the first fracturing fluid that isinduced by formation of a second hydraulic fracture from a treatmentwellbore in the subsurface formation, the second hydraulic fractureformed by a second fracturing fluid in the treatment wellbore;determining a particular pressure signal value of the plurality ofpressure signal values; based on the determined particular pressuresignal value, determining a particular dimension of the second hydraulicfracture formed from the treatment wellbore; determining, based at leastin part on (i) the determined particular dimension, and (ii) a firstpressure signal value of the plurality of pressure signal values that isless than the determined particular pressure signal value, a firstintermediate dimension of the second hydraulic fracture that is lessthan the determined particular dimension of the second hydraulicfracture; and graphically presenting the determined first intermediatedimension to a user.

In an aspect combinable with the general implementation, the particulardimension includes at least one of a half-length of the second hydraulicfracture, a height of the second hydraulic fracture, or an area of thesecond hydraulic fracture.

Another aspect combinable with any of the previous aspects furtherincludes identifying a first fracture stage group of the treatmentwellbore that includes a set of hydraulic fractures formed from thetreatment wellbore.

In another aspect combinable with any of the previous aspects, the setof hydraulic fractures including the second hydraulic fracture formedfrom the treatment wellbore.

Another aspect combinable with any of the previous aspects furtherincludes minimizing an error inequality that includes a ratio of thedetermined fluid pressure and the determined particular pressure signalvalue.

Another aspect combinable with any of the previous aspects furtherincludes determining, based on the minimized error inequality, a commondimension of each of the hydraulic fractures in the set of hydraulicfractures formed from the treatment wellbore.

Another aspect combinable with any of the previous aspects furtherincludes determining, based on the minimized error inequality, adimension of the first hydraulic fracture and a dimension between thefirst hydraulic fracture and the set of hydraulic fractures formed fromthe treatment wellbore.

In another aspect combinable with any of the previous aspects, the errorinequality includes a penalty function.

Another aspect combinable with any of the previous aspects furtherincludes determining the particular dimension of the second hydraulicfracture formed from the treatment wellbore based at least in part on adeviation of the particular dimension from the common dimension of eachof the hydraulic fractures in the set of hydraulic fractures formed fromthe treatment wellbore.

In another aspect combinable with any of the previous aspects,determining the particular dimension of the second hydraulic fractureformed from the treatment wellbore further includes perturbing aplurality of values of the particular dimension of the second hydraulicfracture as a function of the dimension of the first hydraulic fractureand the dimension between the first hydraulic fracture and the set ofhydraulic fractures formed from the treatment wellbore.

In another aspect combinable with any of the previous aspects,perturbing the plurality of values of the particular dimension of thesecond hydraulic fracture includes determining an optimal value of theparticular dimension based on a numerical model that includes: theplurality of values of the particular dimension of the second hydraulicfracture; the plurality of values of the dimension of the firsthydraulic fracture; and the dimension between the first hydraulicfracture and the set of hydraulic fractures formed from the treatmentwellbore.

The computer-implemented method of any one of the previous claims,wherein the intermediate dimension includes at least one of anintermediate half-length of the second hydraulic fracture, anintermediate height of the second hydraulic fracture, or an intermediatearea of the second hydraulic fracture.

Another aspect combinable with any of the previous aspects furtherincludes determining, based at least in part on (i) the determinedparticular dimension, and (ii) a second pressure signal value of theplurality of pressure signal values that is less than the determinedparticular pressure signal value, a second intermediate dimension of thesecond hydraulic fracture that is less than the determined particulardimension of the second hydraulic fracture.

Another aspect combinable with any of the previous aspects furtherincludes generating a fracture growth curve of the second hydraulicfracture based on the first and second intermediate dimensions and thedetermined particular dimension.

Another aspect combinable with any of the previous aspects furtherincludes generating a fracture growth curve of the second hydraulicfracture based on a continuum that includes the first and secondintermediate dimensions and the determined particular dimension.

Another aspect combinable with any of the previous aspects furtherincludes determining additional intermediate dimensions of the secondhydraulic fracture based on the generated fracture growth curve.

Another aspect combinable with any of the previous aspects furtherincludes graphically presenting the generated fracture growth curve to auser.

Another aspect combinable with any of the previous aspects furtherincludes determining, based on the generated fracture growth curve, atleast one hydraulic fracturing operation action.

Another aspect combinable with any of the previous aspects furtherincludes graphically presenting the at least one hydraulic fracturingoperation action to a user.

In another aspect combinable with any of the previous aspects, the atleast one hydraulic fracturing operation action includes at least oneof: an action that adjusts a viscosity of the second fracturing fluidpumped to the treatment wellbore; an action that adjusts a proppantconcentration in the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a pumping rate of the second fracturingfluid pumped to the treatment wellbore; or an action that adjusts adiversion schedule for the treatment wellbore.

In another aspect combinable with any of the previous aspects, the atleast one hydraulic fracturing operation action includes at least oneof: an action that adjusts a viscosity of a third fracturing fluidrelative to a viscosity of the second fracturing fluid pumped to thetreatment wellbore; an action that adjusts a proppant concentration ofthe third fracturing fluid relative to a proppant concentration of thesecond fracturing fluid pumped to the treatment wellbore; an action thatadjusts a pumping rate of the third fracturing fluid relative to apumping rate of the second fracturing fluid pumped to the treatmentwellbore; or an action that adjusts a diversion schedule for a thirdwellbore relative to a diversion schedule for the treatment wellbore.

In another aspect combinable with any of the previous aspects, theparticular pressure signal value corresponds to at least one of: ashut-in time instant of the treatment wellbore upon a cessation ofpumping of the second fracturing fluid into the treatment wellbore; or amaximum pressure of the second fracturing fluid pumped into thetreatment wellbore.

Another aspect combinable with any of the previous aspects furtherincludes normalizing the dimension between the first hydraulic fractureand the set of hydraulic fractures formed from the treatment wellbore inthe numerical model to at least one dimensionless value that representsdimension of the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore.

In another aspect combinable with any of the previous aspects, thedimensionless value is associated with an estimated aspect ratio of thesecond hydraulic fracture that includes a ratio of a height of thesecond hydraulic fracture to a half-length of the second hydraulicfracture, an estimated overlap percentage between the first hydraulicfracture and the second hydraulic fracture, and an estimated offsetbetween the first hydraulic fracture and the second hydraulic fracture.

In another aspect combinable with any of the previous aspects, thenumerical model includes an N-dimensional hypercube.

In another aspect combinable with any of the previous aspects, theN-dimensional hypercube includes a poromechanical model that describes aporomechanical interaction of the second hydraulic fracture and thefirst hydraulic fracture.

Implementations of a hydraulic fracturing modeling system according tothe present disclosure may include one, some, or all of the followingfeatures. For example, implementations may more accurately determinehydraulic fracture dimensions and generate a hydraulic fracture growthcurve, thereby informing a fracture treatment operator about one or moreeffects of particular treatment parameters. Further, implementations maymore accurately determine a proppant area of a hydraulic fracture,thereby increasing the accuracy for hydrocarbon production predictionsfor treated wells.

Implementations of a hydraulic fracturing modeling system according tothe present disclosure may include a system of one or more computersthat can be configured to perform particular actions by virtue of havingsoftware, firmware, hardware, or a combination of them installed on thesystem that in operation causes or cause the system to perform theactions. One or more computer programs can be configured to performparticular actions by virtue of including instructions that, whenexecuted by data processing apparatus, cause the apparatus to performthe actions.

The details of one or more implementations of the subject matterdescribed in this disclosure are set forth in the accompanying drawingsand the description below. Other features, aspects, and advantages ofthe subject matter will become apparent from the description, thedrawings, and the claims.

BRIEF DESCRIPTION OF DRAWINGS

FIGS. 1A-1C are schematic illustrations of an example implementation ofa hydraulic fracturing modeling system within a hydraulic fracturingsystem.

FIG. 2 is a schematic diagram of a structured data processing systemthat implements the hydraulic fracturing modeling system.

FIG. 3 is a flowchart that describes an example method for determining ahydraulic fracture growth curve with a hydraulic fracturing modelingsystem.

FIGS. 4A-4C are schematic illustrations of a monitor wellbore and atreatment wellbore with multiple hydraulic fractures within a hydraulicfracturing stage.

FIG. 4D illustrates a graph that shows results of minimizing the errorspace for multiple fracturing stages of a treatment wellbore of thehydraulic fracturing system.

FIG. 5 is a graphical representation of fracture growth curve generatedby the hydraulic fracturing modeling system during a hydraulicfracturing operation.

FIG. 6 is a graphical representation of a leak-off curve generated bythe hydraulic fracturing modeling system during a hydraulic fracturingoperation.

FIG. 7 is a schematic illustration of a range of proppant-filled area ofa hydraulic fracture determined by the hydraulic fracturing modelingsystem.

FIG. 8 is a graphical representation of a decline in dimensions of amonitor fracture from fracture completion due to leak-off.

FIG. 9 shows a graph of a parameterized decline function overlaid onmultiple treatment times of treatment fracture stages that a monitorfracture observes.

FIG. 10 shows a graph of a step-wise monitor dimension decline based onstage set and trend-line towards the converged monitor dimension todetermine propped fracture dimensions of a monitor fracture.

DETAILED DESCRIPTION

FIGS. 1A-1C are schematic illustrations of an example implementation ofa hydraulic fracturing modeling system 120 (a structured data processingsystem) within a hydraulic fracturing system 100. As shown, system 100includes a monitor wellbore 108 that is formed from a terranean surface102 to a subterranean zone 104 located below the terranean surface 102.The monitor wellbore 108, generally, includes a plug 122 or other fluidbarrier positioned in the wellbore 108, and a pressure sensor 114. Inthis example, the pressure sensor 114 is located at or near a wellheadon the monitor wellbore 108 but in alternate implementations, thepressure sensor 114 may be positioned within the monitor wellbore 108below the terranean surface 102. Generally, according to the presentdisclosure, the monitor wellbore 108 may be used to measure pressurevariations in a fluid contained in the wellbore 108 and/or one or morehydraulic fractures 110 formed from the monitor wellbore 108 that areinduced by a hydraulic fracturing fluid pumped into a treatment wellbore106 to form one or more hydraulic fractures 112 formed from thetreatment wellbore 106. Such induced pressure variations, as explainedmore fully below, may be used to determine a fracture growth curve andother information regarding the hydraulic fractures 112.

The monitor wellbore 108 shown in FIG. 1A includes vertical andhorizontal sections, as well as a radiussed section that connects thevertical and horizontal portions. Generally, and in alternativeimplementations, the wellbore 108 can include horizontal, vertical(e.g., only vertical), slant, curved, and other types of wellboregeometries and orientations. The wellbore 108 may include a casing (notshown) that is cemented or otherwise secured to the wellbore wall todefine a borehole in the inner volume of the casing. In alternativeimplementations, the wellbore 108 can be uncased or include uncasedsections. Perforations (not specifically labeled) can be formed in thecasing to allow fracturing fluids and/or other materials to flow intothe wellbore 108. Perforations can be formed using shape charges, aperforating gun, and/or other tools. Although illustrated as generallyvertical portions and generally horizontal portions, such parts of thewellbore 108 may deviate from exactly vertical and exactly horizontal(e.g., relative to the terranean surface 102) depending on the formationtechniques of the wellbore 108, type of rock formation in thesubterranean formation 104, and other factors. Generally, the presentdisclosure contemplates all conventional and novel techniques forforming the wellbore 108 from the surface 102 into the subterraneanformation 104.

The treatment wellbore 106 shown in FIG. 1A includes vertical andhorizontal sections, as well as a radiussed section that connects thevertical and horizontal portions. Generally, and in alternativeimplementations, the wellbore 106 can include horizontal, vertical(e.g., only vertical), slant, curved, and other types of wellboregeometries and orientations. The treatment wellbore 106 may include acasing (not shown) that is cemented or otherwise secured to the wellborewall to define a borehole in the inner volume of the casing. Inalternative implementations, the wellbore 106 can be uncased or includeuncased sections. Perforations (not specifically labeled) can be formedin the casing to allow fracturing fluids and/or other materials to flowinto the wellbore 106. Perforations can be formed using shape charges, aperforating gun, and/or other tools. Although illustrated as generallyvertical portions and generally horizontal portions, such parts of thewellbore 106 may deviate from exactly vertical and exactly horizontal(e.g., relative to the terranean surface 102) depending on the formationtechniques of the wellbore 106, type of rock formation in thesubterranean formation 104, and other factors. Generally, the presentdisclosure contemplates all conventional and novel techniques forforming the wellbore 106 from the surface 102 into the subterraneanformation 104. Generally, according to the present disclosure, thetreatment wellbore 106 is used to form one or more hydraulic fractures112 that can produce or enhance production of hydrocarbons or otherfluids in the subterranean zone 104. A hydraulic fracturing fluid usedto form such fractures 112, during formation of the fractures 112, mayinduce pressure variations in a fluid contained in the monitor wellbore108, which may be used to determine a fracture growth curve and otherinformation regarding the hydraulic fractures 112.

Although a single monitor wellbore 108 and a single treatment wellbore106 are shown in FIGS. 1A-1C, the present disclosure contemplates thatthe system 100 may include multiple (e.g., more than 2) wellbores, anyof which may be assigned as a “monitor” wellbore or a “treatment”wellbore. For example, in some aspects, there may be multiple (e.g., 10or more) wellbores formed into the subterranean zone 104, with a singlewellbore assigned to be the monitor wellbore and the remaining wellboresassigned to be treatment wellbores. Alternatively, there may be multiplemonitor wellbore and multiple treatment wellbores within a set ofwellbores formed into the subterranean zone. Further, in some aspects,one or more wellbores in a set of wellbores formed into the subterraneanzone 104 may be initially designated as monitor wellbores while one ormore other wellbores may be designated as treatment wellbores. Suchinitial designations, according to the present disclosure, may beadjusted over time such that wellbores initially designated monitorwellbores may be re-designated as treatment wellbores while wellboresinitially designated treatment wellbores may be re-designated as monitorwellbores.

The example hydraulic fracturing system 100 includes a hydraulicfracturing liquid circulation system 118 that is fluidly coupled to thetreatment wellbore 106. In some aspects, the hydraulic fracturing liquidcirculation system 118, which includes one or more pumps 116, is fluidlycoupled to the subterranean formation 104 (which could include a singleformation, multiple formations or portions of a formation) through aworking string (not shown). Generally, the hydraulic fracturing liquidcirculation system 118 can be deployed in any suitable environment, forexample, via skid equipment, a marine vessel, sub-sea deployedequipment, or other types of equipment and include hoses, tubes, fluidtanks or reservoirs, pumps, valves, and/or other suitable structures andequipment arranged to circulate a hydraulic fracturing liquid throughthe treatment wellbore 106 and into the subterranean formation 104 togenerate the one or more fractures 112. The working string is positionedto communicate the hydraulic fracturing liquid into the treatmentwellbore 106 and can include coiled tubing, sectioned pipe, and/or otherstructures that communicate fluid through the wellbore 106. The workingstring can also include flow control devices, bypass valves, ports, andor other tools or well devices that control the flow of fracturing fluidfrom the interior of the working string into the subterranean formation104.

Although labeled as a terranean surface 102, this surface may be anyappropriate surface on Earth (or other planet) from which drilling andcompletion equipment may be staged to recover hydrocarbons from asubterranean zone. For example, in some aspects, the surface 102 mayrepresent a body of water, such as a sea, gulf, ocean, lake, orotherwise. In some aspects, all are part of a drilling and completionsystem, including hydraulic fracturing system 100, may be staged on thebody of water or on a floor of the body of water (e.g., ocean or gulffloor). Thus, references to terranean surface 102 includes reference tobodies of water, terranean surfaces under bodies of water, as well asland locations.

Subterranean formation 104 includes one or more rock or geologicformations that bear hydrocarbons (e.g., oil, gas) or other fluids(e.g., water) to be produced to the terranean surface 102. For example,the rock or geologic formations can be shale, sandstone, or other typeof rock, typically, that may be hydraulically fractured to produce orenhance production of such hydrocarbons or other fluids.

As shown specifically in FIG. 1C, the monitor fractures 110 emanatingfrom the monitor wellbore 108 and the treatment fractures 112 emanatingfrom the treatment wellbore 106 may extend past each other (e.g.,overlap in one or two dimensions) when formed. In some aspects, dataabout the location of such fractures 110 and 112 and their respectivewellbores 108 and 106, such as locations of the wellbores, distancesbetween the wellbores (e.g., in three dimensions) depth of horizontalportions of the wellbores, and locations of the hydraulic fracturesinitiated from the wellbores (e.g., based on perforation locationsformed in the wellbores), among other information. In some aspects, suchinformation (along with the monitored, induced pressure variations in afluid in the one or more monitor wellbores) may be used to helpdetermine one or more dimensions (e.g., fracture length, fracturehalf-length, fracture height, fracture area) of the hydraulic fractures112.

FIG. 2 is a schematic diagram of a computing system that implements thehydraulic fracturing modeling system 120 (structured data processingsystem) shown in FIGS. 1A-1C. Generally, the hydraulic fracturingmodeling system 120 includes a processor-based control system operableto implement one or more operations described in the present disclosure.As shown in FIG. 2, pressure signal values 142 may be received at thehydraulic fracturing modeling system 120 from the pressure sensor 114that is fluidly coupled to or in the monitor wellbore 108. The pressuresignal values 142, in some aspects, may represent pressure variations ina fluid that is enclosed or contained in the monitor wellbore 108 and/orthe hydraulic fractures 110 that are induced by a hydraulic fracturingfluid being used to form hydraulic fractures 112 from the treatmentwellbore 106.

The hydraulic fracturing modeling system 120 may be any computing deviceoperable to receive, transmit, process, and store any appropriate dataassociated with operations described in the present disclosure. Theillustrated hydraulic fracturing modeling system 120 includes hydraulicfracturing modeling application 130. The application 130 is any type ofapplication that allows the hydraulic fracturing modeling system 120 torequest and view content on the hydraulic fracturing modeling system120. In some implementations, the application 130 can be and/or includea web browser. In some implementations, the application 130 can useparameters, metadata, and other information received at launch to accessa particular set of data associated with the hydraulic fracturingmodeling system 120. Further, although illustrated as a singleapplication 130, the application 130 may be implemented as multipleapplications in the hydraulic fracturing modeling system 120.

The illustrated hydraulic fracturing modeling system 120 furtherincludes an interface 136, a processor 134, and a memory 132. Theinterface 136 is used by the hydraulic fracturing modeling system 120for communicating with other systems in a distributedenvironment—including, for example, the pressure sensor 114—that may beconnected to a network. Generally, the interface 136 comprises logicencoded in software and/or hardware in a suitable combination andoperable to communicate with, for instance, the pressure sensor 114, anetwork, and/or other computing devices. More specifically, theinterface 136 may comprise software supporting one or more communicationprotocols associated with communications such that a network orinterface's hardware is operable to communicate physical signals withinand outside of the hydraulic fracturing modeling system 120.

Regardless of the particular implementation, “software” may includecomputer-readable instructions, firmware, wired or programmed hardware,or any combination thereof on a tangible medium (transitory ornon-transitory, as appropriate) operable when executed to perform atleast the processes and operations described herein. Indeed, eachsoftware component may be fully or partially written or described in anyappropriate computer language including C, C++, Java, Visual Basic,ABAP, assembler, Perl, Python, .net, Matlab, any suitable version of4GL, as well as others. While portions of the software illustrated inFIG. 2 are shown as individual modules that implement the variousfeatures and functionality through various objects, methods, or otherprocesses, the software may instead include a number of sub-modules,third party services, components, libraries, and such, as appropriate.Conversely, the features and functionality of various components can becombined into single components as appropriate.

The processor 134 executes instructions and manipulates data to performthe operations of the hydraulic fracturing modeling system 120. Theprocessor 134 may be a central processing unit (CPU), a blade, anapplication specific integrated circuit (ASIC), a field-programmablegate array (FPGA), or another suitable component. Generally, theprocessor 134 executes instructions and manipulates data to perform theoperations of the hydraulic fracturing modeling system 120.

Although illustrated as a single memory 132 in FIG. 2, two or morememories may be used according to particular needs, desires, orparticular implementations of the hydraulic fracturing modeling system120. In some implementations, the memory 132 is an in-memory database.While memory 132 is illustrated as an integral component of thehydraulic fracturing modeling system 120, in some implementations, thememory 132 can be external to the hydraulic fracturing modeling system120. The memory 132 may include any memory or database module and maytake the form of volatile or non-volatile memory including, withoutlimitation, magnetic media, optical media, random access memory (RAM),read-only memory (ROM), removable media, or any other suitable local orremote memory component. The memory 132 may store various objects ordata, including classes, frameworks, applications, backup data, businessobjects, jobs, web pages, web page templates, database tables,repositories storing business and/or dynamic information, and any otherappropriate information including any parameters, variables, algorithms,instructions, rules, constraints, or references thereto associated withthe purposes of the hydraulic fracturing modeling system 120.

The illustrated hydraulic fracturing modeling system 120 is intended toencompass any computing device such as a desktop computer,laptop/notebook computer, wireless data port, smart phone, smart watch,wearable computing device, personal data assistant (PDA), tabletcomputing device, one or more processors within these devices, or anyother suitable processing device. For example, the hydraulic fracturingmodeling system 120 may comprise a computer that includes an inputdevice, such as a keypad, touch screen, or other device that can acceptuser information, and an output device that conveys informationassociated with the operation of the hydraulic fracturing modelingsystem 120 itself, including digital data, visual information, or a GUI.

As illustrated in FIG. 2, the memory 132 stores structured data,including one or more poromechanical models 138. In some aspects, aporomechanical model 138 may be in the form of an N-dimensionalhypercube that describes poromechanical interactions between thehydraulic fractures 110 and the hydraulic fractures 112. Theporomechanical interactions may be identified using pressure signalsmeasured by the pressure sensor 114 of a fluid contained in the monitorwellbore 108 or the hydraulic fractures 110. The poromechanicalinteractions may also be identified using one or more pressure sensorsor other components that measure a pressure of a hydraulic fracturingfluid used to form the hydraulic fractures 112 from the treatmentwellbore 106. In certain embodiments, the pressure signals include apressure versus time curve of the pressure signal. Pressure-inducedporomechanic signals may be identified in the pressure versus time curveand the pressure-induced poromechanic signals may be used to assess oneor more parameters (e.g., geometry) of the hydraulic fractures 112. Asused herein, a “pressure-induced poromechanic signal” refers to arecordable change in pressure of a first fluid in direct fluidcommunication with a pressure sensor (e.g., pressure gauge) where therecordable change in pressure is caused by a change in stress on a solidin a subsurface formation that is in contact with a second fluid (e.g.,a hydrocarbon fluid), which is in direct fluid communication with thefirst fluid. The change in stress of the solid may be caused by a thirdfluid used in a hydraulic stimulation process (e.g., a hydraulicfracturing process) in a treatment wellbore 106 in proximity to (e.g.,adjacent) the observation (monitoring) wellbore with the third fluid notbeing in direct fluid communication with the second fluid.

For example, a pressure-induced poromechanic signal may occur in thepressure sensor 114 attached to the wellhead of the monitor wellbore108, where at least one stage of that monitor wellbore 108 has alreadybeen hydraulically fractured to create the fractures 110 (assumed, forthis example, to be part of a common fracturing stage), when theadjacent treatment wellbore 106 undergoes hydraulic stimulation. Aparticular hydraulic fracture 112 emanating from the treatment wellbore106 may grow in proximity to the fracture 110 but these fractures do notintersect. No fluid from the hydraulic fracturing process in thetreatment wellbore 106 contacts any fluid in the hydraulic fractures 110and no measurable pressure change in the fluid in the hydraulicfractures 110 is caused by advective or diffusive mass transport relatedto the hydraulic fracturing process in the treatment wellbore 106. Thus,the interaction of the fluids in the hydraulic fracture 112 with fluidsin the subsurface matrix does not result in a recordable pressure changein the fluids in the fracture 110 that can be measured by the pressuresensor 114. The change in stress on a rock (in the subterranean zone104) in contact with the fluids in the fracture 112, however, may causea change in pressure in the fluids in the fracture 110, which can bemeasured as a pressure-induced poromechanic signal in the pressuresensor 114.

Poromechanic signals may be present in traditional pressure measurementstaken in the monitor wellbore 108 while fracturing the treatmentwellbore 106. For example, if a newly formed hydraulic fracture 112overlaps or grows in proximity to a particular hydraulic fracture 110 influid communication with the pressure sensor 114 in the monitor wellbore108, one or more poromechanic signals may be present. However,poromechanic signals may be smaller in nature than a direct fluidcommunication signal (e.g., a direct pressure signal induced by directfluid communication such as a direct fracture hit or fluid connectivitythrough a high permeability fault). Poromechanic signals may alsomanifest over a different time scale than direct fluid communicationsignals. Thus, poromechanic signals are often overlooked, unnoticed, ordisregarded as data drift or error in the pressure sensor 114. However,such signals may be used, at least in part, to determine a fracturegrowth curve and other associated fracture dimensions of the hydraulicfractures 112 that emanate from the treatment wellbore 106.

Turning briefly to FIG. 4A, this figures shows a schematic view of thesystem 100 in which a hydraulic fracturing stage 113 is shown to includemultiple hydraulic fractures 112 from the treatment wellbore 106. Asingle hydraulic fracture 110 (which also is part of a hydraulicfracturing stage with multiple fractures, not shown) is illustrated inthis figure. In some aspects, the poromechanical model 138 comprisesnumerical model that calculates a gain, K, for each combination of amonitor stage fracture (fracture 110), i, and a fractured stage, j,according to:K _(ij)=ƒ({tilde under (D)} _(m) ^(i) ,{tilde under (D)} _(ƒ) ^(j),{right arrow over (X)} _(ij))  Eq. (1)

where D_(m) and D_(ƒ) are the dimensions of the hydraulic fracture 110and the fractured stage 113, respectively, and i ranges over the numberof hydraulic fractures 112 in a particular stage of hydraulic fractures112 from the monitor wellbore 108, and j ranges over the number ofstages 113 that are completed. The vector, X_(ij)€R³, connects aperforation location, j, for a particular stage 113 of hydraulicfractures 112 and a perforation location, i, of the hydraulic fracture110. This vector is approximately known since each relative position ofall perforation locations (in both the monitor wellbore 108 and thetreatment wellbore 106) is specified. Thus, only a combination of whichlocations (in the monitor wellbore 108 and the treatment wellbore 106)need be solved for to determine the gain, K, of that combination, ij.Further, the tilde underneath the dimensions, D, indicates that a“dimension” may include a set of geometry attributes such as fracturehalf-length on both sides of a wellbore, fracture height, fractureazimuth, vertical asymmetry with respect to wellbore, and fractureshape, rather than a single “dimension.”

The function, ƒ, in Eq. 1 relies on the poromechanical model 138 as wellas one or more normalized dimensions (that may be part of the model 138or separate from the model 138). For example, the normalized dimensionsinclude a dimension between the hydraulic fracture 110 and the set ofhydraulic fractures 112 formed from the treatment wellbore 106. Thenormalized dimension includes a dimensionless value that is associatedwith an aspect ratio of a height of the hydraulic fracture 112 to ahalf-length of the hydraulic fracture 112. The dimensionless value mayalso be associated with an estimated overlap percentage between thehydraulic fracture 110 and the hydraulic fracture 112 (e.g., as shown inFIG. 1C). The dimensionless value may also be associated with anestimated offset between the hydraulic fracture 110 and the secondhydraulic fracture 112.

Besides the spatial information and the geometry information of thefractures 110 and 112, there is also sequence/time data in the numericalmodel. In some examples, this timing information is captured by theBoolean function:

$\begin{matrix}{\tau_{ij} = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu} t_{i}^{start}} < t_{j}^{frac} < t_{i}^{stop}} \\0 & {{for}\mspace{14mu}{other}}\end{matrix} \right.} & {{Eq}.\mspace{11mu} 2}\end{matrix}$

where τ_(ij) equals one only when a fracture j is being treated(fracturing time: t_(j) ^(frac)) during the “lifetime” of monitor, I,bounded by its start, t_(i) ^(start) and stop time t_(i) ^(stop). Whenthe monitor is “not live” during the fracturing of a stage, τ_(ij) equalzero.

The gain, K, may be obtained through pressure signals received from thepressure sensor 114 as well as a net pressure measured for fracturingthe hydraulic fracture stage 113. For example, from field measurements(such as those described), the measured gain value is obtained, whichequals a ratio of the ΔP_(ij) observed by monitor, i, during fracturingof stage, j (i.e., an observation) and the net pressure, P^(j) _(net),measured for fracturing of stage, j. Thus, the calculated gain, K_(ij),can be set equal to the measured gain value:

$\begin{matrix}{{K_{ij}\tau_{ij}} = {\frac{\Delta\; P_{ij}}{P_{net}^{j}} + {\omega.}}} & {{Eq}.\mspace{11mu} 3}\end{matrix}$

Here, unknown values on the right hand side of Eq. 3 (the measured gainvalue) are monitor fracture (110) dimensions, D′m, treatment fracture(112) dimensions, D^(i) _(ƒ), and the perforation location vector. Thefluctuation value, ω, captures all errors arising from model assumptions(in K_(ij)) and field measurement variability

$\left( {{in}\mspace{20mu}\frac{\Delta\; P_{ij}}{P_{net}^{j}}} \right).$In some aspects, the value of ω may determine the accuracy of measuredgain. In some aspects, an assumption that ω may be small (e.g.,approaching zero) and therefor negligible may be made.

FIG. 3 is a flowchart that describes an example method 300 fordetermining a hydraulic fracture growth curve with the hydraulicfracturing modeling system 120, using the poromechanical model 138. Forexample, method 300 may begin at step 302, which includes identifyingporomechanic pressure signals from a pressure sensor fluidly coupled toa monitor wellbore that includes at least one hydraulic fractureporomechanically coupled to an adjacent treatment wellbore.

Method 300 may continue at step 304, which includes determining, basedon the poromechanic pressure values, a fracture stage dimension of thehydraulic fractures of a particular fracture stage from the treatmentwellbore. For example, in step 304, a determination of a common, orrepresentative, dimension of all of the hydraulic fractures in aparticular fracture stage is made. Turning briefly to FIG. 4B, forexample, the determined common dimension (e.g., fracture half-length)115 is calculated and assumed (in this step) to be the same value foreach of the hydraulic fractures 112 that emanate from the treatmentwellbore 106.

In some aspects, for example, the fracture stage 113 may be assumed tohave multiple hydraulic fractures 112 that were formed through similar,if not identical, fracture treatments (e.g., based on pump rates andpressure, time of fracture operation, viscosity and proppant mixture offracturing fluid, and otherwise). Thus, this assumption leads to theassumption that each hydraulic fracture 112 in stage 113 may havesimilar, final dimensions. In step 304 of method 300, the fracturedimensions of a stage group, n (e.g., stage group 113) is referred to byD^(n) _(G), yielding:{tilde under (D)} _(ƒ) ^(j) ={tilde under (D)} _(G) ^(n(j))  Eq. 4

where n(j) implies that a stage j is uniquely assigned to a single stagegroup n. If the number of stage groups, N, is smaller than the number offractured stages J (N<J), then the number of degrees of freedom from ourfractured stage dimensions D^(i) _(ƒ) is reduced by a fraction of J/N.

In some aspects, the solution space may be constrained by constrainingthe vector, X_(i), to a finite number of discrete values (e.g., ratherthan a continuous variable) governed by the (discrete) cluster locationsin the monitor stage and the treatment stage. Also, there may belimiting factors for the ranges of the fracture dimensions, for examplea known formation barrier (e.g., between the subterranean zone 104 andthe terranean surface 102) that constrains a height of the fractures112.

In some aspects, there may be fractured stage dimensions that remainunconstrained (or very poorly conditioned). For example, in cases wherethere is a large amount of time between stage fracturing, or betweenfracturing of the monitor wellbore 108 and the treatment wellbore 106,or a large distance between the fractures 110 and the fractures 112,these fractured stages may be removed from the system and will not bemapped. This also applies to monitor stages. For example, if a monitorstage didn't make any or an insufficient number of observations, themonitor dimensions for that particular monitor stage may remainunresolved.

In step 304, assuming ω=0 from Eq. 3 and including the degree of freedomreduction from Eq. 4 yields:

$\begin{matrix}{{{K_{ij}\left( {{\underset{\sim}{D}}_{f}^{j}, \sim} \right)}\tau_{ij}} = \left. \frac{\Delta\; P_{ij}}{P_{net}^{j}}\rightarrow{{{K_{ij}\left( {{\underset{\sim}{D}}_{G}^{n{(j)}}, \sim} \right)}\tau_{ij}} \approx {\frac{\Delta\; P_{ij}}{P_{net}^{j}}.}} \right.} & {{Eq}.\mspace{11mu} 5}\end{matrix}$

This system may be solved by minimizing an overall error posed by theinequality of Eq. 5 by minimizing an error function, £, arising from theinequality:

$\begin{matrix}{{E = {\min{\sum\limits_{i}{\sum\limits_{j}{\omega_{ij}\epsilon_{ij}\mspace{14mu}{with}}}}}}{\epsilon_{ij} = {{\epsilon\left( {{K_{ij}\tau_{ij}},{\Delta\; P_{ij}\text{/}P_{net}^{j}}} \right)}.}}} & {{Eq}.\mspace{11mu} 6}\end{matrix}$

In Eq. 6, w_(ij) is the weight factor of the individual error. In someaspects, minimizing the error space may be algorithmically executed orexecuted algorithmically and with human intelligence (e.g., to analyzeand interpret the impact of the error ε_(ij) on the overall solution.The penalty factor w_(ij) may be used to set the weight factor for acertain error (or observation). This value can be automatically set byalgorithms or can be overwritten by an analyst based on experience.Turning briefly to FIG. 4D, this graph illustrates the results ofminimizing the error space for multiple fracturing stages (1 . . . j) ofthe treatment wellbore 106. As shown, there may be a maximum and minimumdetermined value of the gain, K_(ij), for each stage that results fromthe error minimization.

The final global solution resulting from step 304 includes adetermination of the monitor fracture dimension (e.g., of fracture 110),D^(i) _(m); the common or representative dimension 115 of the treatmentfracture stage 113 (that included hydraulic fractures 112), and thevector, X_(ij), that represents the relative position of the fractures110 and 112 in the stage 113 (e.g., as constrained by the discreteperforation locations).

Method 300 may continue at step 306, which includes determining, basedon the fracture stage dimension, a dimension of a particular hydraulicfracture from the treatment wellbore that is within the particularfracture stage. For example, in step 306, a determination of aparticular dimension (e.g., a maximum dimension) of a particularhydraulic fracture 112 (or each hydraulic fracture 112 if repeated foreach particular fracture 112 in the stage 113) is made. Turning brieflyto FIG. 4C, for example, each fracture 112 may have a differentdimension (e.g., fracture half-length or otherwise) as shown. Step 306determines the particular, unique dimension of the hydraulic fracture112.

In some aspects, step 306 includes a perturbation method to determinehow much a dimension of an individual hydraulic fracture 112 deviatesfrom the common or representative dimension that is determined in step304. For example, step 306 includes determining how much an individualfracture dimension, D^(i) _(ƒ), actually deviates from the overalldimension of the stage, D^(n) _(G)(j). In some aspects, this may bedetermined by selecting the best observation(s) (i, j) for a stage j.For that stage j, the equality from Eq. 3 (assuming zero error), byperturbing D^(i) _(ƒ), while leaving the monitor fracture dimensions,D^(i) _(m), unaltered. Further, the relative perforation locations,vector X_(ij), may only change by changing the cluster number of thetreatment wellbore fracturing stage that is completed (e.g., the clusternumber of the monitor wellbore 108 stays fixed).

In some aspects, not all fracture dimension attributes of the stagescompleted on the treatment wellbore are perturbed. For example, in someaspects, only the fracture half-lengths, FHL_(ij) is used as anindependent variable. Thus, in some examples, a fracture height iseither kept constant or scaled along with the changing fracturehalf-length of the particular hydraulic fracture 112. In some aspects,the fracture cluster may be varied as well, in order to satisfy theequality. Thus, step 306 may determine a localized (e.g., for aparticular hydraulic fracture 112) dimension.

Method 300 may continue at step 308, which includes generating afracture growth curve based on the poromechanic pressure values. Forexample, once a local solution in step 306 is determined, furtherdimensions (e.g., fracture half-length) of the particular hydraulicfracture 112 with respect to time may be determined to obtain thefracture growth curve. For example, the observed pressure in the monitorwellbore 110 (e.g., from the pressure sensor 114) relative to fractureoperation time during a treatment in the treatment wellbore 106,

${\frac{\Delta\; P_{ij}}{\Delta\; P_{net}^{j}}(t)},$is transformed into a time-dependent fracture dimension, FHL_(ij)(t) byapplying a transfer function, B. B, in some aspects, is non-linear as afunction of time. Accordingly:

$\begin{matrix}{{B\left( {{FHLij}(t)} \right)} = {{\frac{\Delta\; P_{ij}}{\Delta\; P_{net}^{j}}(t)} \approx {{K_{ij}(t)}.}}} & {{Eq}.\mspace{11mu} 7}\end{matrix}$

Thus, the transfer function can be estimated by the gain, K_(ij). Insome aspects, this transfer function is implemented as a lookup table140 stored in the memory 132 of the hydraulic fracturing modeling system120. Once multiple values of the FHL_(ij) (t) are determined, thefracture growth curve can be plotted, such as is shown in FIG. 5. FIG. 5is a graphical representation 500 of fracture growth curve 512 generatedby the hydraulic fracturing modeling system during a hydraulicfracturing operation. As shown, the fracture dimension (in this case,fracture half-length) is represented on the y-axis 504, while time(e.g., pump time operation of pump 116) is represented on the x-axis502. As shown, different portions of the fracture growth curve 512 maybe overlaid, or sectioned, according to particular fracture treatmentparameters. For example, as shown, three different treatment parameters“A”, “B”, . . . , “n” are overlaid in order to relate the fracturegrowth curve to the actual fracture treatment of the treatment wellbore110. These treatment parameters can be, for example, a first fracturingfluid type that was used in a first half of the pump time and a secondfracturing fluid type that was used for the second half of the pumptime. Treatment parameters can also involve the treatment pressure,proppant concentration, fluid rate, moment that a diverter is dropped,different types of proppant (e.g., mesh size of the sand). Thus, changesin the fracture growth curve can be related to a treatment parameterchange, if any.

Method 300 may include further operations and steps as well. Forexample, in some aspects, the generated fracture growth curve may bepresented to the treatment operator, as well as recommendations based onthe curve. For instance, recommendations may include adjusting one ormore parameters of the current hydraulic fracturing operation in thetreatment wellbore 106 or a future hydraulic fracturing operation (e.g.,in treatment wellbore 106 or another wellbore).

Implementations according to the present disclosure may also includecomputer-implemented methods, systems and apparatus for determining aproppant area of a hydraulic fracture. Proppant (e.g., sand or otherparticle) may be mixed with a hydraulic fracturing fluid to holdfractures open after a hydraulic fracturing treatment, e.g., in thetreatment well 106. Thus, in some aspects, the proppant bears a closureweight of the fracture once the hydraulic fracturing fluid leaks off(thereby removing the fluid pressure from bearing the weight of theclosure). By holding open the fracture, the proppant may create an openconduit for production of hydrocarbon fluid from the subterranean zone104 to the wellbore 106. In some aspects, however, proppant in thehydraulic fracturing fluid may not extend or reach to a maximumdimension (e.g., fracture half length, fracture length, fracture height)of the hydraulic fracture. Thus, even though the hydraulic fracture mayinitially have a certain maximum dimension, without proppant extendingto that maximum dimension, the fracture will start to close off at thatdimension over time (e.g., once the hydraulic fracturing fluid pressureis removed from the fracture).

In some aspects, techniques for determining a proppant area of ahydraulic fracture may build on the method 300 described herein thatresults in a determination of a hydraulic fracture growth curve. Forexample, at least a portion of method 300 may be used to determine aproppant area of the hydraulic fracture based on a measured leak-offpressure curve relative to pump run time. For example, FIG. 6 shows agraphical representation 600 of a leak-off curve generated by thehydraulic fracturing modeling system 120 during a hydraulic fracturingoperation. As shown, graph 600 includes a y-axis 604 that defines afluid pressure measured by pressure sensor 114 at the monitor wellbore108 and an x-axis 602 that is defined by time, and more specifically, arun-time of the hydraulic fracturing fluid pump 116 (or pumps 116). Bymeasuring the pressure signal values over the run time of the pump 116(e.g., during a hydraulic fracturing operation in the treatment wellbore106 to generate a particular hydraulic fracture 112), a pressure vs.time (or “leak-off”) curve 604 is generated as shown. As shown in FIG.6, once the fracture treatment ends (e.g., the pumps shut off), thepressure begins to drop and eventually levels off in the “posttreatment” area of the curve 604. The drop in the pressure (e.g., fromthe shut-off time to point 616 on the curve 604) may represent theleaking off of the fracturing fluid before the hydraulic fracture beginsto close down on any proppant embedded in the fracturing fluid and leftbehind in the fracture (e.g., a proppant “landing point”). In someaspects, the proppant landing point may be a proppant landing range overtime, rather than a distinct point in time.

Based on the curve 604, the landing point (or range) of the proppant maybe determined, for example, by transforming the curve 604 into apressure vs. square root of time curve (from a pressure vs. time curve).A derivative of the transformed curve 604 may then be taken to determinea proppant landing time range, e.g., a range in which the fractureclosure pressure switches from being borne by the hydraulic fracturingfluid pressure to being borne by the proppant. For example, as shown onthe curve 604, point 616 represents a beginning time instant of theproppant landing range and point 618 represents an end time instant ofthe proppant landing range. Points 616 and 618 are determined, forexample, based on the derivative of the transformed curve 604.

Once the proppant landing time range is found (as shown by points 616and 618), a “pre-closure” portion of the curve 604 (to the left of point616) and a “post-closure” portion of the curve 604 (to the right ofpoint 618). In order to find the fluid pressure range that correspondsto the time range of the landing range, two back-extrapolated curves(608 and 610, shown in dotted lines) may be determined. Theback-extrapolated curves 610 and 608 start from points 618 and 616,respectively, and intersect the leak-off curve 604 at points 614 and612, respectively. Point 612 on the curve 604, as shown, represents afluid pressure (measured by the pressure sensor 114 at the monitorwellbore 108) that corresponds to the maximum propped dimension (e.g.,fracture half-length or fracture length). Point 614 on the curve 604, asshown, represents a fluid pressure (measured by the pressure sensor 114at the monitor wellbore 108) that corresponds to the minimum proppeddimension (e.g., fracture half-length or fracture length). Turningbriefly to FIG. 7, this figure is a schematic illustration 700 of arange of proppant-filled area of a hydraulic fracture determined by thehydraulic fracturing modeling system 120. For instance, as shown, theminimum propped dimension (e.g., area) is shown as area 706 while themaximum propped dimension (e.g., area) is shown as area 704. Thedimension of the hydraulic fracture (as determined by method 300) isrepresented as area 702 and, as shown, is larger than the propped arearange.

Once points 612 and 614 are determined on the curve 604, these pressurevalues may be used in method 300 to determine a “propped” fracturedimension (e.g., area as derived from fracture half-length) range. Forexample, the pressure values at points 612 and 614 may be inserted asthe “identified poromechanic pressure signals” in step 304. Thus,execution of steps 304 and 306, as described previously, determine twopropped fracture half-lengths (a minimum corresponding to point 614 anda maximum corresponding to point 612). The two determined proppedfracture half-lengths may then be used to calculate two propped fractureareas (e.g., assuming constrained height of the fracture).

Other techniques may be used to determine a propped fracture area of ahydraulic fracture from a monitor wellbore. For example, in someaspects, the determination of propped fracture area of a monitorwellbore according to the previous description, which relies on themethod 300 and the leak-off curve 604, may assume or consider that thefracture dimensions of both the treatment fracture and the monitorfracture are constant over time. For the set of dimensions of thetreatment fracture, D_(ƒ), this may be a valid assumption, since asnap-shot of the hydraulic fracture dimensions at the moment when thisfracture is completed is being determined.

For the set of monitor fracture dimensions, D_(m), however, thisassumption may be too strict, since a monitor stage can be “active”(e.g., change dimension) for an extended period of time, typicallyspanning from a view hours to weeks. During this period, the monitorfracture may lose fluid into the formation (leak-off), which reduces thetotal amount of fluid in the monitor fracture and may result in agradual closure of the fracture on the proppant. Resulting from theleak-off, the effective dimensions of the monitor fracture may typicallyshrink as the monitor fracture goes from a fully fluid-supportedfracture (for example, right after treatment completion) to aproppant-supported fracture.

Therefore, in the limit, the monitor fracture dimensions may converge tothe propped fracture dimensions. For example, as shown in FIG. 8, agraphical representation 800 of a decline in dimensions of a monitorfracture from fracture completion due to leak-off is illustrated. Graph800 includes a y-axis 802 that represents the set of monitor fracturedimensions, D_(m), and an x-axis 804 that represents time. Graph 800includes a curve 806 that represents D_(m) over time startingapproximately at the completion time instant of the monitor fracture.Point 808 on graph 800 represents the dimension of the monitor fracture(e.g., at its largest) at the completion time instant of the monitorfracture. Line 810 represents the asymptotic propped fracture dimensionof the monitor fracture. As shown, curve 806 approaches (converges to)line 810 as time goes to infinity (i.e., much after the monitor fracturehas been completed). This convergence of the monitor dimensions towardsthe propped dimensions may be leveraged.

For example, Equation 1 is rewritten here in slightly different form:K _(ij)=ƒ(D(t _(ƒ))_(m) ^(i) ,D _(ƒ) ^(j) ,{right arrow over (X)}_(ij))  Eq. 1

Here, the dimensions, D, may include a set of geometry attributes suchas fracture half-length on both sides of a wellbore, fracture height,fracture azimuth, vertical asymmetry with respect to wellbore, andfracture shape, rather than a single “dimension.” Also, in this versionof Equation 1, t_(ƒ) is the treatment time of the fractured stage, ƒ.This slightly different form of Equation 1 follows from allowing atransient monitor fracture dimension.

This version of Equation 1 indicates that for each of the successivestages monitored, the monitor dimensions can vary. In some aspects, themonitor dimensions vary by decreasing. In order to match the monitordimensions, some level of constraint may be needed; otherwise therewould be an under-constrained (e.g., non-unique) situation, where themonitor (with its constant decreasing dimensions) observes insufficienttreatment stages to be accurately determined.

In some aspects, there may be several possible types of constraints thatmay be applied, each of which providing a process for determining apropped area of a fracture. For example, a first type of constraint isbased on a functional (parametric) description of the transient monitordimension. As another example, a second type of constraint may rely onan assumption that, over short periods of time, the monitor dimensioncan be assumed to be constant.

In an example aspect associated with the first type of constraint, adecline function may be defined for the decreasing monitor dimensions:D(t _(ƒ))_(m) ^(i) =D(t _(∞))_(m) ^(i)+[D(t ₀)_(m) ^(i) −D(t(t _(∞))_(m)^(i)]exp(−αΔt)  Eq. 8,

Where α is leak-off rate, D(t_(∞))_(m) ^(i) is the propped fracturedimensions, and D(t₀)_(m) ^(i) is the hydraulic fracture hydraulicdimensions. Here, Δt equals the time interval between the completion ofthe monitor stage time instant (t_(m)) and the completion of stage, ƒ,(t_(ƒ)). In other words, Δt=t_(ƒ)−t_(m). In alternative implementations,a different decline function, or another power law function may bedefined for the decreasing monitor dimensions.

These parameters may be added to the set of degrees of freedom (degreeof freedoms) that is being solved for according to step 304 of method300. As described previously, completion of step 304 provides the finalglobal solution that includes a determination of the monitor fracturedimension (e.g., of fracture 110), D^(i) _(m); the common orrepresentative dimension 115 of the treatment fracture stage 113 (thatincluded hydraulic fractures 112), and the vector, X_(ij), thatrepresents the relative position of the fractures 110 and 112 in thestage 113 (e.g., as constrained by the discrete perforation locations).Thus, in this aspect of determining a propped area of a hydraulicfracture according to the first type of constraint, two monitor fracturedimensions (initial hydraulic fracture dimension of the monitor fractureand final propped dimensions of the monitor fracture) are simultaneouslysolved. Thus, the solution of this system may directly provide thedesired propped fracture dimensions.

In another example aspect, the second type of constraint may rely on anassumption that, over short periods of time, the monitor dimension canbe assumed to be constant. This may enable the definition of sets ofobserved stages, which are completed in a relatively short time period,and during which completion, the monitor dimension maybe approximated tobe constant. This second type of constraint may yield a two-stepprocess. For example, as a first sub-step, FIG. 9 shows a graph 900 of aparameterized decline function overlaid on multiple treatment times oftreatment fracture stages that a monitor fracture observes. Graph 900includes a y-axis 902 that represents the set of dimensions of a monitorfracture stage at a treatment time of the treatment fracture stage.Graph 900 also includes an x-axis 904 that represents time, e.g., fromthe monitor fracture stage. Curve 906 represents the dimensions of themonitor fracture, D_(m), over time starting approximately at thecompletion time instant of the monitor fracture stage. As shown, points910 represent the dimensions of the monitor fracture, D_(m), atparticular time observed fracture stages 912 of the treatment well. Aswith FIG. 8, point 908 represents the dimension of the monitor fracture(e.g., at its largest) at the completion time instant of the monitorfracture. Line 914 represents the asymptotic propped fracture dimensionof the monitor fracture. As shown, curve 906 approaches (converges to)line 914 as time goes to infinity (i.e., much after the monitor fracturehas been completed).

Solving for the system according to step 304 of method 300, for thissecond type of constraint, may not directly yield the desired proppedfracture dimensions. Thus, as a second sub-step, a trend line may befitted through the mapped points 910 (t_(j)D(t_(j))_(m) ^(i)) of curve906. For example, FIG. 10 shows a graph 1000 of a step-wise monitordimension decline based on stage set and trend-line towards theconverged monitor dimension to determine propped fracture dimensions ofa monitor fracture. Graph 1000 includes a y-axis 1002 that representsthe set of dimensions of a monitor fracture stage at a treatment time ofthe treatment fracture stage. Graph 1000 also includes an x-axis 1004that represents time, e.g., from the monitor fracture stage. Curve 1006represents the trend line of the dimensions of the monitor fracture,D_(m), over time starting approximately at the completion time instantof the monitor fracture stage. Point 1008 represents the dimension ofthe monitor fracture (e.g., at its largest) at the completion timeinstant of the monitor fracture. Line 1014 represents the asymptoticpropped fracture dimension of the monitor fracture. As shown, curve 1006approaches (converges to) line 1014 as time goes to infinity (i.e., muchafter the monitor fracture has been completed).

Regarding the first sub-step, as shown in graph 1000, three points areused to define the trend line 1006. The first point 1008 represents thedimension of the monitor fracture as of the completion time, t₀, of thisfracture. The dimensions of the monitor fracture at this time instantmust be the largest estimate of the propped fracture dimensions of themonitor fracture, i.e., the propped dimensions cannot be larger than thelargest estimate of the monitor fracture dimensions. The second point1013 is shown as having dimensions less than those of point 1008, e.g.,due to leakoff, and represents the dimensions of the monitor fracture ata time of a first stage set completion, t₁, past the completion time,t₀. In other words, the dimensions at point 1013 represent the monitorfracture dimensions when the first stage set of treatment fracturesobserved by the monitor fracture is completed. The third point 1015 isshown as having dimensions less than those of points 1008 and 1013,e.g., due to leakoff, and represents the dimensions of the monitorfracture at a time of a second stage set completion, t₂, past thecompletion time, t₀. In other words, the dimensions at point 1015represent the monitor fracture dimensions when the second stage set oftreatment fractures observed by the monitor fracture is completed.

Regarding the second sub-step, the trend line 1006, given the threepoints 1008, 1013, and 1015, may then be extrapolated to determine thedecreasing trend of the propped monitor fracture dimensions. In someaspects, a regression curve may be fit to the three points 1008, 1013,and 1015. In some aspects, more points (e.g., more than three) may bedetermined (e.g., for third, fourth, and additional stage sets of thetreatment fracture). The trend curve 1006, therefore, may be regressionfitted to more than three points as well.

The features described can be implemented in digital electroniccircuitry, or in computer hardware, firmware, software, or incombinations of them. The apparatus can be implemented in a computerprogram product tangibly embodied in an information carrier, forexample, in a machine-readable storage device for execution by aprogrammable processor; and method steps can be performed by aprogrammable processor executing a program of instructions to performfunctions of the described implementations by operating on input dataand generating output. The described features can be implementedadvantageously in one or more computer programs that are executable on aprogrammable system including at least one programmable processorcoupled to receive data and instructions from, and to transmit data andinstructions to, a data storage system, at least one input device, andat least one output device. A computer program is a set of instructionsthat can be used, directly or indirectly, in a computer to perform acertain activity or bring about a certain result. A computer program canbe written in any form of programming language, including compiled orinterpreted languages, and it can be deployed in any form, including asa stand-alone program or as a module, component, subroutine, or otherunit suitable for use in a computing environment.

Suitable processors for the execution of a program of instructionsinclude, by way of example, both general and special purposemicroprocessors, and the sole processor or one of multiple processors ofany kind of computer. Generally, a processor will receive instructionsand data from a read-only memory or a random access memory or both. Theessential elements of a computer are a processor for executinginstructions and one or more memories for storing instructions and data.Generally, a computer will also include, or be operatively coupled tocommunicate with, one or more mass storage devices for storing datafiles; such devices include magnetic disks, such as internal hard disksand removable disks; magneto-optical disks; and optical disks. Storagedevices suitable for tangibly embodying computer program instructionsand data include all forms of non-volatile memory, including by way ofexample semiconductor memory devices, such as EPROM, EEPROM, and flashmemory devices; magnetic disks such as internal hard disks and removabledisks; magneto-optical disks; and CD-ROM and DVD-ROM disks. Theprocessor and the memory can be supplemented by, or incorporated in,ASICs (application-specific integrated circuits).

To provide for interaction with a user, the features can be implementedon a computer having a display device such as a CRT (cathode ray tube)or LCD (liquid crystal display) monitor for displaying information tothe user and a keyboard and a pointing device such as a mouse or atrackball by which the user can provide input to the computer.Additionally, such activities can be implemented via touchscreenflat-panel displays and other appropriate mechanisms.

The features can be implemented in a control system that includes aback-end component, such as a data server, or that includes a middlewarecomponent, such as an application server or an Internet server, or thatincludes a front-end component, such as a client computer having agraphical user interface or an Internet browser, or any combination ofthem. The components of the system can be connected by any form ormedium of digital data communication such as a communication network.Examples of communication networks include a local area network (“LAN”),a wide area network (“WAN”), peer-to-peer networks (having ad-hoc orstatic members), grid computing infrastructures, and the Internet.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinventions or of what may be claimed, but rather as descriptions offeatures specific to particular implementations of particularinventions. Certain features that are described in this specification inthe context of separate implementations can also be implemented incombination in a single implementation. Conversely, various featuresthat are described in the context of a single implementation can also beimplemented in multiple implementations separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made without departingfrom the spirit and scope of the disclosure. For example, exampleoperations, methods, or processes described herein may include moresteps or fewer steps than those described. Further, the steps in suchexample operations, methods, or processes may be performed in differentsuccessions than that described or illustrated in the figures.Accordingly, other implementations are within the scope of the followingclaims.

What is claimed is:
 1. A structured data processing system fordetermining a hydraulic fracture dimension, comprising: one or morehardware processors; a memory in communication with the one or morehardware processors, the memory storing a data structure and anexecution environment, the data structure storing data that comprises aplurality of pressure signal values of a first fracturing fluid in amonitor wellbore, formed from a terranean surface into a subsurfaceformation, the first fracturing fluid in direct fluid communication witha first hydraulic fracture formed from the monitor wellbore into thesubsurface formation, each of the plurality of pressure signal valuescomprising a pressure change in the first fracturing fluid that isinduced by formation of a second hydraulic fracture from a treatmentwellbore in the subsurface formation, the second hydraulic fractureformed by pumping a second fracturing fluid in the treatment wellbore,the execution environment comprising: a fracture growth solverconfigured to perform operations comprising: determining a particularpressure signal value of the plurality of pressure signal values; basedon the determined particular pressure signal value, determining aparticular value of a dimension of the second hydraulic fracture formedfrom the treatment wellbore, the dimension of the second hydraulicfracture comprising a geometric attribute of the second hydraulicfracture; and determining, based at least in part on (i) the determinedparticular value of the dimension, and (ii) a first pressure signalvalue of the plurality of pressure signal values that is less than thedetermined particular pressure signal value, a first intermediate valueof the dimension of the second hydraulic fracture that is less than thedetermined particular value of the dimension of the second hydraulicfracture, the first intermediate value being between an initial value ofthe dimension of the second hydraulic fracture and a value of thedimension of the second hydraulic fracture at a pump shut-off of thepumping of the second fracturing fluid in the treatment wellbore; a userinterface module that generates a user interface that renders one ormore graphical representations of the determined first intermediatevalue of the dimension of the second hydraulic fracture; and atransmission module that transmits, over one or more communicationprotocols and to a computing device, data that represents the one ormore graphical representations.
 2. The system of claim 1, wherein thegeometric attribute of the second hydraulic fracture comprises at leastone of a value of a half-length of the second hydraulic fracture, avalue of a height of the second hydraulic fracture, or a value of anarea of the second hydraulic fracture.
 3. The system of claim 1, whereinthe fracture growth solver is configured to perform operations furthercomprising: identifying a first fracture stage group of the treatmentwellbore that comprises a set of hydraulic fractures formed from thetreatment wellbore, the set of hydraulic fractures including the secondhydraulic fracture formed from the treatment wellbore; minimizing anerror inequality that comprises a ratio of the determined fluid pressureand the determined particular pressure signal value; determining, basedon the minimized error inequality, a common dimension of each of thehydraulic fractures in the set of hydraulic fractures formed from thetreatment wellbore; and determining, based on the minimized errorinequality, a dimension of the first hydraulic fracture and a dimensionbetween the first hydraulic fracture and the set of hydraulic fracturesformed from the treatment wellbore.
 4. The system of claim 3, whereinthe error inequality comprises a penalty function.
 5. The system ofclaim 3, wherein the fracture growth solver is configured to performoperations further comprising determining the particular value of thedimension of the second hydraulic fracture formed from the treatmentwellbore based at least in part on a deviation of the particular valueof the dimension from the common dimension of each of the hydraulicfractures in the set of hydraulic fractures formed from the treatmentwellbore.
 6. The system of claim 5, wherein the operation of determiningthe particular value of the dimension of the second hydraulic fractureformed from the treatment wellbore further comprises perturbing aplurality of values of the dimension of the second hydraulic fracture asa function of the dimension of the first hydraulic fracture and thedimension between the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore.
 7. The system of claim 6,wherein the operation of perturbing the plurality of values of thedimension of the second hydraulic fracture comprises determining anoptimal value of the dimension based on a numerical model thatcomprises: the plurality of values of the dimension of the secondhydraulic fracture; the plurality of values of the dimension of thefirst hydraulic fracture; and the dimension between the first hydraulicfracture and the set of hydraulic fractures formed from the treatmentwellbore.
 8. The system of claim 1, wherein the first intermediate valueof the dimension comprises at least one of an intermediate value of ahalf-length of the second hydraulic fracture, an intermediate value of aheight of the second hydraulic fracture, or an intermediate value of anarea of the second hydraulic fracture.
 9. The system of claim 1, whereinthe fracture growth solver is configured to perform operations furthercomprising: determining, based at least in part on (i) the determinedparticular value of the dimension, and (ii) a second pressure signalvalue of the plurality of pressure signal values that is less than thedetermined particular pressure signal value, a second intermediate valueof the dimension of the second hydraulic fracture that is less than thedetermined particular value of the dimension of the second hydraulicfracture, the second intermediate value of the dimension being betweenthe initial value of the dimension of the second hydraulic fracture andthe value of the dimension of the second hydraulic fracture at the pumpshut-off of the pumping of the second fracturing fluid in the treatmentwellbore; and generating a fracture growth curve of the second hydraulicfracture based on the first and second intermediate values of thedimension and the determined particular value of the dimension.
 10. Thesystem of claim 9, wherein the fracture growth solver is configured toperform operations further comprising: generating the fracture growthcurve of the second hydraulic fracture based on a continuum thatcomprises the first and second intermediate values of the dimension andthe determined particular value of the dimension; and determiningadditional intermediate values of the dimension of the second hydraulicfracture based on the generated fracture growth curve.
 11. The system ofclaim 9, wherein the fracture growth solver is configured to performoperations further comprising graphically presenting the generatedfracture growth curve to a user.
 12. The system of claim 9, wherein thefracture growth solver is configured to perform operations furthercomprising determining, based on the generated fracture growth curve, atleast one hydraulic fracturing operation action, the user interfacemodule is configured to generate a user interface that renders one ormore graphical representations of the at least one hydraulic fracturingoperation action, and the transmission module is configured to transmit,over the one or more communication protocols and to the computingdevice, data that represents the one or more graphical representationsof the at least one hydraulic fracturing operation action.
 13. Thesystem of claim 12, wherein the at least one hydraulic fracturingoperation action comprises at least one of: an action that adjusts aviscosity of the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a proppant concentration in the secondfracturing fluid pumped to the treatment wellbore; an action thatadjusts a pumping rate of the second fracturing fluid pumped to thetreatment wellbore; an action that adjusts a diversion schedule for thetreatment wellbore; an action that adjusts a viscosity of a thirdfracturing fluid relative to a viscosity of the second fracturing fluidpumped to the treatment wellbore; an action that adjusts a proppantconcentration of the third fracturing fluid relative to a proppantconcentration of the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a pumping rate of the third fracturingfluid relative to a pumping rate of the second fracturing fluid pumpedto the treatment wellbore; or an action that adjusts a diversionschedule for a third wellbore relative to a diversion schedule for thetreatment wellbore.
 14. The system of claim 1, wherein the particularpressure signal value corresponds to at least one of: a shut-in timeinstant of the treatment wellbore upon a cessation of pumping of thesecond fracturing fluid into the treatment wellbore; or a maximumpressure of the second fracturing fluid pumped into the treatmentwellbore.
 15. The system of claim 5, wherein the fracture growth solveris configured to perform operations further comprising normalizing thedimension between the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore in the numerical model toat least one dimensionless value that represents a dimension of thefirst hydraulic fracture and the set of hydraulic fractures formed fromthe treatment wellbore, and the dimensionless value is associated withan estimated aspect ratio of the second hydraulic fracture thatcomprises a ratio of a height of the second hydraulic fracture to ahalf-length of the second hydraulic fracture, an estimated overlappercentage between the first hydraulic fracture and the second hydraulicfracture, and an estimated offset between the first hydraulic fractureand the second hydraulic fracture.
 16. A computer-implemented method fordetermining a hydraulic fracture dimension, comprising: identifying,with one or more hardware processors, a plurality of pressure signalvalues of a first fracturing fluid in a monitor wellbore, formed from aterranean surface into a subsurface formation, the first fracturingfluid in direct fluid communication with a first hydraulic fractureformed from the monitor wellbore into the subsurface formation, each ofthe plurality of pressure signal values comprising a pressure change inthe first fracturing fluid that is induced by formation of a secondhydraulic fracture from a treatment wellbore in the subsurfaceformation, the second hydraulic fracture formed by a second fracturingfluid in the treatment wellbore; determining, with the one or morehardware processors, a particular pressure signal value of the pluralityof pressure signal values; based on the determined particular pressuresignal value, determining, with the one or more hardware processors, aparticular value of a dimension of the second hydraulic fracture formedfrom the treatment wellbore, the dimension of the second hydraulicfracture comprising a geometric attribute of the second hydraulicfracture; determining, with the one or more hardware processors, basedat least in part on (i) the determined particular value of thedimension, and (ii) a first pressure signal value of the plurality ofpressure signal values that is less than the determined particularpressure signal value, a first intermediate value of the dimension ofthe second hydraulic fracture that is less than the determinedparticular value of the dimension of the second hydraulic fracture, thefirst intermediate value being between an initial value of the dimensionof the second hydraulic fracture and a value of the dimension of thesecond hydraulic fracture at a pump shut-off of the pumping of thesecond fracturing fluid in the treatment wellbore; and graphicallypresenting, with the one or more hardware processors, the determinedfirst intermediate value of the dimension to a user.
 17. Thecomputer-implemented method of claim 16, wherein the geometric attributeof the second hydraulic fracture comprises at least one of a value of ahalf-length of the second hydraulic fracture, a value of a height of thesecond hydraulic fracture, or a value of an area of the second hydraulicfracture.
 18. The computer-implemented method of claim 16, furthercomprising: identifying, with the one or more hardware processors, afirst fracture stage group of the treatment wellbore that comprises aset of hydraulic fractures formed from the treatment wellbore, the setof hydraulic fractures including the second hydraulic fracture formedfrom the treatment wellbore; minimizing, with the one or more hardwareprocessors, an error inequality that comprises a ratio of the determinedfluid pressure and the determined particular pressure signal value; anddetermining, with the one or more hardware processors, based on theminimized error inequality, a common dimension of each of the hydraulicfractures in the set of hydraulic fractures formed from the treatmentwellbore; and determining, with the one or more hardware processors,based on the minimized error inequality, a dimension of the firsthydraulic fracture and a dimension between the first hydraulic fractureand the set of hydraulic fractures formed from the treatment wellbore.19. The computer-implemented method of claim 18, wherein the errorinequality comprises a penalty function.
 20. The computer-implementedmethod of claim 18, further comprising determining, with the one or morehardware processors, the particular value of the dimension of the secondhydraulic fracture formed from the treatment wellbore based at least inpart on a deviation of the particular value of the dimension from thecommon dimension of each of the hydraulic fractures in the set ofhydraulic fractures formed from the treatment wellbore.
 21. Thecomputer-implemented method of claim 20, wherein determining theparticular value of the dimension of the second hydraulic fractureformed from the treatment wellbore further comprises perturbing aplurality of values of the dimension of the second hydraulic fracture asa function of the dimension of the first hydraulic fracture and thedimension between the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore.
 22. Thecomputer-implemented method of claim 21, wherein perturbing theplurality of values of the dimension of the second hydraulic fracturecomprises determining an optimal value of the dimension based on anumerical model that comprises: the plurality of values of the dimensionof the second hydraulic fracture; the plurality of values of thedimension of the first hydraulic fracture; and the dimension between thefirst hydraulic fracture and the set of hydraulic fractures formed fromthe treatment wellbore.
 23. The computer-implemented method of claim 16,wherein the intermediate value of the dimension comprises at least oneof an intermediate value of a half-length of the second hydraulicfracture, an intermediate value of a height of the second hydraulicfracture, or an intermediate value of an area of the second hydraulicfracture.
 24. The computer-implemented method of claim 16, furthercomprising: determining, with the one or more hardware processors, basedat least in part on (i) the determined particular value of thedimension, and (ii) a second pressure signal value of the plurality ofpressure signal values that is less than the determined particularpressure signal value, a second intermediate value of the dimension ofthe second hydraulic fracture that is less than the determinedparticular value of the dimension of the second hydraulic fracture, thesecond intermediate value of the dimension being between the initialvalue of the dimension of the second hydraulic fracture and the value ofthe dimension of the second hydraulic fracture at the pump shut-off ofthe pumping of the second fracturing fluid in the treatment wellbore;and generating, with the one or more hardware processors, a fracturegrowth curve of the second hydraulic fracture based on the first andsecond intermediate values of the dimension and the determinedparticular value of the dimension.
 25. The computer-implemented methodof claim 24, further comprising: generating, with the one or morehardware processors, the fracture growth curve of the second hydraulicfracture based on a continuum that comprises the first and secondintermediate value of the dimension and the determined particular valueof the dimension; and determining, with the one or more hardwareprocessors, additional intermediate values of the dimension of thesecond hydraulic fracture based on the generated fracture growth curve.26. The computer-implemented method of claim 24, further comprising:determining, with the one or more hardware processors, based on thegenerated fracture growth curve, at least one hydraulic fracturingoperation action; and graphically presenting the at least one hydraulicfracturing operation action to a user.
 27. The computer-implementedmethod of claim 26, wherein the at least one hydraulic fracturingoperation action comprises at least one of: an action that adjusts aviscosity of the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a proppant concentration in the secondfracturing fluid pumped to the treatment wellbore; an action thatadjusts a pumping rate of the second fracturing fluid pumped to thetreatment wellbore; an action that adjusts a diversion schedule for thetreatment wellbore; an action that adjusts a viscosity of a thirdfracturing fluid relative to a viscosity of the second fracturing fluidpumped to the treatment wellbore; an action that adjusts a proppantconcentration of the third fracturing fluid relative to a proppantconcentration of the second fracturing fluid pumped to the treatmentwellbore; an action that adjusts a pumping rate of the third fracturingfluid relative to a pumping rate of the second fracturing fluid pumpedto the treatment wellbore; or an action that adjusts a diversionschedule for a third wellbore relative to a diversion schedule for thetreatment wellbore.
 28. The computer-implemented method of claim 16,wherein the particular pressure signal value corresponds to at least oneof: a shut-in time instant of the treatment wellbore upon a cessation ofpumping of the second fracturing fluid into the treatment wellbore; or amaximum pressure of the second fracturing fluid pumped into thetreatment wellbore.
 29. The computer-implemented method of claim 21,further comprising normalizing, with the one or more hardwareprocessors, the dimension between the first hydraulic fracture and theset of hydraulic fractures formed from the treatment wellbore in thenumerical model to at least one dimensionless value that represents adimension of the first hydraulic fracture and the set of hydraulicfractures formed from the treatment wellbore, and the dimensionlessvalue is associated with an estimated aspect ratio of the secondhydraulic fracture that comprises a ratio of a height of the secondhydraulic fracture to a half-length of the second hydraulic fracture, anestimated overlap percentage between the first hydraulic fracture andthe second hydraulic fracture, and an estimated offset between the firsthydraulic fracture and the second hydraulic fracture.
 30. Thecomputer-implemented method of claim 16, wherein the numerical modelincludes an N-dimensional hypercube, and the N-dimensional hypercubeincludes a poromechanical model that describes a poromechanicalinteraction of the second hydraulic fracture and the first hydraulicfracture.